Inequalities concerning the \(L^p\)-norm of a polynomial
Let \(K\geq 1\), \(\mu\geq 1\) and let \(p(z)=a_0+a_\mu z^\mu+\cdots+a_nz= ^n\) be a complex polynomial of degree \(n\) not vanishing in the disk \(|z|<K\). The authors show that for \(q\geq 1\) one has \(||p'||_q\leq c||p||_q,\) where \(||\cdot||_q\) denotes the \(L_p\) norm on \(|z|=1\) and \(c\) is an explicitly given constant, which depends on \(n\), \(\mu\), \(K\), \(p(0)\) and the coefficient \(a_\mu\) of \(f\). This generalizes a result of de Bruijn who considered the case \(K=\mu=1\).
scientific article; zbMATH DE number 1215751